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Fraction Word Problems

Here are some examples of fraction word problems.The first example is a one-step word problem.

The second example shows how blocks can be used to help illustrate the problem.The third example is a two-step word problem.

Example 1:

Martha spent of her allowance on food and shopping. What fraction of her allowance had sheleft?

Solution:

She had of her allowance left.

Example 2:

of a group of children were girls. If there were 24 girls, how many children were there in the

group?

Solution:

3 units = 24

1 unit = 24 3 = 8

5 units = 5 8 = 40

There were 40 children in the group.


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Proportion Word Problems

Proportion problems are word problems where the items in the question are proportional to eachother. They are two main types of proportional problems:Directly Proportional Problemsand

Inversely Proportional Problems.

Directly Proportional Problems

The question usually will not tell you that the items aredirectly proportional. Instead, it will giveyou the value of two items which are related and then asks you to figure out what will be thevalue of one of the item if the value of the other item changes.

Proportion problems are usually of the form:

Ifx theny. Ifx is changed to a then what will be the value ofy?

For example,

If two pencils cost $1.50, how many pencils can you buy with $9.00?

The main difficulty with this type of question is to figure out which values to divide and whichvalues to multiply.

The following method is helpful:

Change the word problem into the form:


which can then be represented as:

For example,

You can think of the sentence:

If two pencils cost $1.50, how many pencils can you buy with $9.00?as If$1.50 then two pencils.If$9.00 then how many pencils?

Write the proportional relationship:
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Inversely Proportional Problems

Inversely Proportional questions are similar to directly proportional problems, but the differenceis that whenx increasey will decrease and vice versa - which is theinverse proportionrelationship. The most common example of inverse proportion problems would be the more

men on a job the less timetaken for the job to complete

Again, the technique is to change the proportion problems into the form:


and then write the inverse relationship (take note of the "inverse" form):

Example:It takes 4 men 6 hours to repair a road. How long will it take 7 men to do the job if they work atthe same rate?

Step 1: Think of the word problem as:

If 4 then 6. If 7 then how many?

Step 2: Write out the inverse relationship:

Answer: They will take hours.
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Percentage word problems

Before you learn about percentage word problems, reviewFormula for percentageor you can usethe approach that I use here.

Example #1:

A test has 20 questions. If peter gets 80% correct, how many questions did peter missed?The number of correct answers is 80% of 20 or 80/100 20

80/100 20 = 0.80 20 = 16

Recall that 16 is called the percentage. It is the answer you get when you take the percent of a

numberSince the test has 20 questions and he got 16 correct answers, the number of questions he missedis 20 16 = 4

Peter missed 4 questions

Example #2:

In a school, 25 % of the teachers teach basic math. If there are 50 basic math teachers, how manyteachers are there in the school?

I shall help you reason the problem out:When we say that 25 % of the teachers teach basic math, we mean 25% of all teachers in theschool equal number of teachers teaching basic mathSince we don't know how many teachers there are in the school, we replace this with x or a blankHowever, we know that the number of teachers teaching basic or the percentage = 50

Putting it all together, we get the following equation:

25% of ____ = 50 or 25% ___ = 50 or 0.25 ____ = 50

Thus, the question is 0.25 times what gives me 50

A simple division of 50 by 0.25 will get you the answer

50/0.25 = 200

Therefore, we have 200 teachers in the school

In fact, 0.25 200 = 50
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Number Problem involving one unknown

There are several problems which involve relations among known and unknown numbers and can be

put in the form of equations. The equations are generally stated in words and it is for this reason we

refer to these problems as word problems. With the help of equations in one variable, we have already

practiced equations to solve some real life problems.

Step-by-step application of linear equations to solve practical word problems:

1. The sum of two numbers is 25. One of the numbers exceeds the other by 9. Find the numbers.

Solution:

Then the other number = x + 9Let the number be x.Sum of two numbers = 25

According to question, x + x + 9 = 25

2x + 9 = 25 2x = 25 - 9 (transposing 9 to the R.H.S changes to -9)

2x = 16 2x/2 = 16/2 (divide by 2 on both the sides) x = 8

Therefore, x + 9 = 8 + 9 = 17

Therefore, the two numbers are 9 and 16.

2.The difference between the two numbers is 48. The ratio of the two numbers is 7:3. What arethe two numbers?

Solution:Let the common ratio be x.Let the common ratio be x.Their difference = 48According to the question,

7x - 3x = 48 4x = 48 x = 48/4 x = 12

Therefore, 7x = 7 12 = 84

3x = 3 12 = 36


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Consecutive Integer Problems

Consecutive integer problems are word problems that involveconsecutive integers.

Consecutive integers are integers that follow in sequence, each number being 1 more than theprevious number, represented by n, n +1, n + 2, n + 3, ..., where n is any integer.

For example: 23, 24, 25,

The following are common examples of consecutive integer problems.

Example #1:

Find four consecutive even integers such that the sum of the second and fourth integers is oneand a half times the sum of the first and third integers.

Solution #1:

Consecutive Even Integers: n, n + 2, n + 4, n + 6 for even integer nSum of Second and Fourth Integers: (n + 2) + (n + 6) = 2n + 8Sum of First and Third Integers: n + (n + 4) = 2n + 4

Since sum of the second and fourth integers is twice than the sum of the first and third integers, itfollows that:

2n + 8 = 1.5(2n + 4)2n + 8 = 3n + 63n 2n = 8 6n = 2

Four consecutive even integers are 2, 4, 6, and 8

Example #2:

Find three consecutive odd integers such that the sum of the second and third integers is fourtimes the first.

Solution #2:

Consecutive Even Integers: n, n + 2, n + 4 for odd integer nSum of Second and Third Integers: (n + 2) + (n + 4) = 2n + 6Since sum of the second and third integers equals four times the first integer, it follows that:2n + 6 = 4n4n 2n = 62n = 6n = 3

Three consecutive odd integers are 3, 5 and 7
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Age Word Problems

Age problems are algebra word problems that deal with the ages of people currently, in the pastor in the future.

If the problem involves a single person, then it is similar to an Integer Problem. Read theproblem carefully to determine the relationship between the numbers. This is shown in theexample involving a single person.

If the age problem involves the ages of two or more people then using a table would be a goodidea. A table will help you to organize the information and to write the equations. This is shownin the age problems:examples involving more than one person.

Age Problems Involving A Single Person

Example 1:

Five years ago, Johns age was half of the age he will be in 8 years. How old is he now?

Solution:

Step 1: Letxbe Johns age now. Look at the question and put the relevant expressions above it.

Step 2: Write out the equation.

Isolatevariablex

Answer: John is now 18 years old.
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Age Problems Involving More Than One Person

Example 2:

John is twice as old as his friend Peter. Peter is 5 years older than Alice. In 5 years, John will be

three times as old as Alice. How old is Peter now?

Solution:

Step 1: Set up a table.

age now age in 5 yrs

John

Peter

Alice

Step 2: Fill in the table with information given in the question.

John is twice as old as his friend Peter. Peter is 5 years older than Alice. In 5 years, John will bethree times as old as Alice. How old is Peter now?

Letxbe Peters age now. Add 5 to get the ages in 5 yrs.

age now age in 5 yrs

John 2x 2x + 5

Peter x x + 5

Alice x5 x5 + 5

Write the new relationship in an equation using the ages in 5 yrs.

In 5 years, John will be three times as old as Alice.

2x + 5 = 3(x5 + 5)

2x + 5 = 3x

Isolatevariablex

x = 5

Answer: Peter is now 5 years old.
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Average Word Problems

There are three main types of algebra average word problems commonly encountered in schoolor in tests like the SAT:Average (Arithmetic Mean),Weighted AverageandAverage Speed.

Average (Arithmetic Mean)

The average (arithmetic mean) uses the formula:

The formula can also be written as

Example:

The average (arithmetic mean) of a list of 6 numbers is 20. If we remove one of the numbers, theaverage of the remaining numbers is 15. What is the number that was removed?

Solution:

Step 1: The removed number could be obtained by difference between the sum of original 6

numbers and the sum of remaining 5 numbers i.e.

sum of original 6 numberssum of remaining 5 numbers

Step 2: Using the formula

sum of original 6 numbers = 20 6 = 120sum of remaining 5 numbers = 15 5 = 75

Step 3: Using the formula from step 1

Number removed = sum of original 6 numberssum of remaining 5 numbers

12075 = 45

Answer: The number removed is 45.
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Weighted Mean Word Problem

Another type of average problem involves the weighted average - which is the average of two ormore terms that do not all have the same number of members. To find the weighted term,multiply each term by its weighting factor, which is the number of times each term occurs.

The formula for weighted average is:

Example:

A class of 25 students took a science test. 10 students had an average (arithmetic mean) score of80. The other students had an average score of 60. What is the average score of the whole class?

Solution:

Step 1: To get the sum of weighted terms, multiply each average by the number of students thathad that average and then sum them up.

80 10 + 60 15 = 800 + 900 = 1700

Step 2: Total number of terms = Total number of students = 25

Step 3: Using the formula

Answer: The average score of the whole class is 68.

Be careful! You will get the wrong answer if you add the two average scores and divide theanswer by two.


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Mixture Word Problems

Mixture problems are word problems where items or quantities of different values are mixedtogether.

Sometimes different liquids are mixed together changing the concentration of the mixture asshown inexample 1,example 2andexample 3.

Sometimes quantities of different costs are mixed together as shown inexample 4.

We recommend using a table to organize your information for mixture problems. Using a tableallows you to think of one number at a time instead of trying to handle the whole mixtureproblem at once.

We will show you how it is done by the following examples of mixture problems:

Adding to the SolutionRemoving from the SolutionReplacing the SolutionMixing Quantities of Different Costs

Adding To The Solution

Mixture Problems: Example 1:

John has 20 ounces of a 20% of salt solution, How much salt should he add to make it a 25%

solution?

Solution:

Step 1: Set up a table for salt.

original added result

concentration

amount


John has 20 ounces of a 20% of salt solution. How much salt should he add to make it a 25%solution?

The salt added is 100% salt, which is 1 in decimal.Change all thepercentto decimals

Letx = amount of salt added. The result would be 20 +x.
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concentration 0.2 1 0.25

amount 20 x 20 +x

Step 3: Multiply down each column.


concentration 0.2 1 0.25

amount 20 x 20 +x

multiply 0.2 20 1 x 0.25(20 +x)

Step 4: original + added = result

0.2 20 + 1 x = 0.25(20 +x)4 +x = 5 + 0.25x

Isolatevariablexx0.25x = 540.75x = 1

Answer: He should add ounces of salt.

Removing From The Solution

Mixture Problems: Example 2:

John has 20 ounces of a 20% of salt solution. How much water should he evaporate to make it a30% solution?

Solution:

Step 1: Set up a table for water. The water is removed from the original.

original removed result

concentration

amount

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Step 1: Set up a table for alcohol. The alcohol is replaced i.e. removed and added.

original removed added result

concentration

amount


A tank has a capacity of 10 gallons. When it is full, it contains 15% alcohol. How many gallonsmust be replaced by an 80% alcohol solution to give 10 gallons of 70% solution?

Change all thepercentto decimals.

Letx = amount of alcohol solution replaced.


concentration 0.15 0.15 0.8 0.7

amount 10 x x 10

Step 3: Multiply down each column.


concentration 0.15 0.15 0.8 0.7

amount 10 x x 10

multiply 0.15 10 0.15 x 0.8 x 0.7 10

Step 4: Since the alcohol solution is replaced, we need to subtract and add.

originalremoved + added = result0.15 100.15 x + 0.8 x = 0.7 101.50.15x + 0.8x = 7

Isolatevariablex0.8x0.15x = 71.50.65x = 5.5

Answer: 8.46 gallons of alcohol solution needs to be replaced.
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Algebra Motion Problems

In this lesson, we will learn how to solve algebra word problems that involve motion.

Motion problems are based on the formula

d = rt

where d= distance, r= rate and t= time.

When solving motion problems, a sketch isoften helpful and a table can be used fororganizing information.

Example:

John and Philip who live 14 miles apart startat noon to walk toward each other at rates of3 mph and 4 mph respectively. In how manyhours will they meet?

Solution:

Letx = time walked.

r t d

John 3 x 3xPhilip 4 x 4x

3x + 4x = 147x = 14x = 2

They will meet in 2 hours.

Example:

In still water, Peters boat goes 4 times asfast as the current in the river. He takes a 15-mile trip up the river and returns in 4 hours.

Find the rate of the current.

Solution:

Letx = rate of the current.

r t d

down river 4x +x 15 / 5x 15

up river 4x -x 15 / 3x 15

The rate of the current is 2 mph.


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Interest Word Problems

Interest Problems are word problems that use the formula for Simple Interest. There is alsoanother type of interest called Compound Interest but we will not be discussing that in this

lesson.

The formula for Simple Interest is:

i = prt

i is the interest generated.p is the principal amount that is either invested or owedris the rate at which the interest is paidtis the time that the principal amount is either invested or owed

This type of word problem is not difficult. Just remember the formula and make sure you plug inthe right values. The rate is usually given inpercent, which you will need to change to a decimalvalue.

Example 1:

John wants to have an interest income of$3,000 a year. How much must he invest forone year at 8%?

Solution:

Step 1: Write down the formula

i = prt

Step 2: Plug in the values

3000 =p 0.08 13000 = 0.08p

p = 37,500

Answer: He must invest $37,500

Example 2:

Jane owes the bank some money at 4% peryear. After half a year, she paid $45 asinterest. How much money does she owe the

bank?

Solution:

Step 1: Write down the formula

i = prt

Step 2: Plug in the values

45 = 0.02pp = 2250

Answer: She owes $2,250
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Work Word Problems

Work Problems are word problems that involve different people doing work together but atdifferent rates. If the people were working at the same rate then we would use theInversely

Proportional Methodinstead.

In this lesson, we will learn

how to solve work problems that involve two persons how to solve work problems that involve more than two persons how to solve work problems that involve pipes filling up a tank The formula for Work Problems that involvetwo personsis

This formula can be extended formore than two persons. It can also be used in problems

that involvepipes filling up a tank.

"Work" Problems: Two Persons

Example 1: Peter can mow the lawn in 40 minutes and John can mow the lawn in 60 minutes. How

long will it take for them to mow the lawn together? Solution: Step 1: Assignvariables: Letx = time to mow lawn together Step 2: Use the formula:

Step 3: Solve the equation TheLCMof 40 and 60 is 120

Multiply both sides with 120

Answer: The time taken for both of them to mow the lawn together is 24 minutes.
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Work Problems: More than Two Persons

Example 1:

Jane, Paul and Peter can finish painting the fence in 2 hours. If Jane does the job alone she can

finish it in 5 hours. If Paul does the job alone he can finish it in 6 hours. How long will it take forPeter to finish the job alone?

Solution:

Step 1: Assignvariables:

Letx = time taken by Peter

Step 2: Use the formula:

Step 3: Solve the equation

Multiply both sides with 30x

Answer: The time taken for Peter to paint the fence alone is hours.
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Work Problems: Pipes Filling up a Tank

Example 1:

A tank can be filled by pipe A in 3 hours and by pipe B in 5 hours. When the tank is full, it can

be drained by pipe C in 4 hours. if the tank is initially empty and all three pipes are open, howmany hours will it take to fill up the tank?

Solution:

Step 1: Assignvariables:

Letx = time taken to fill up the tank

Step 2: Use the formula:

Since pipe C drains the water it is subtracted.

Step 3: Solve the equation

TheLCMof 3, 4 and 5 is 60

Multiply both sides with 60

Answer: The time taken to fill the tank is hours.
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Andres Soriano College

Mangagoy, Bislig City


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OPERATIONS OF REAL NUMBERS

Operations refer to addition, subtraction, multiplication and division.

ADDITION/SUBTRACTION OF REAL NUMBERS

a) Two real numbers with the same signs, add and use the common sign

b) Two real numbers with different signs, subtract and use the sign of the larger number

EXAMPLES:

1) -5 + (-3) = -8 (same signs, add and use the common sign)2) 3 + (-7) = -4 (different signs, subtract and use the sign of larger number)3) -14 + 23 = 9 (different signs, subtract and use the sign of larger number)

4) 28 + (-12) = 16 (different signs, subtract and use the sign of larger number)5) -42 - 21 = -43 (same signs, add and use the common sign)

can also be written as -42 + (-21) = -436) -3/5 + 2/5 = -1/5 (different signs, subtract and use the sign of larger number)7) -4.5 + (-3.6) + (-1.1) = -9.2 (same signs, add and use the common sign)

8) -12 - 13 + 42 - 11 + 16 + (-12) + 15(-12-13-11-12) + (42+16+15) (group the neg numbers and pos numbers)

-48 + 7325

NOTE: when there are grouping symbols such as the absolute value and opposites, you mustsimplify those first and then add/subtract.

9) 8 - (-6) (first take the opposite of -6, then add/subtract)8 + 6 = 14

10) 4.5 + (-6.0) - (-2.3) - 12 + 2.6 - (-2.1)4.5 + (-6.0) + 2.3 - 12 + 2.6 + 2.1 (simplify the opposites)(4.5 + 2.3 + 2.6 + 2.1) + (-6.0 -12) (group like terms)

11.5 + (-18)-6.5

11) |-2| + (-6) + 12 - (-3)2 + (-6) + 12 + 3 (simplify absolute value and opposites)

1112) -(-6) - 26 - |-14| + 34 (simplify absolute value and opposites)

6 - 26 - 14 + 34 (note: - |-14| 14)(6 + 34) + (-26 - 14)

40 + (-40)0


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MULTIPLICATION/DIVISION OF REAL NUMBERS

a) do the indicated operation

b) two positive numbers or two negative numbers, then answer is a positive

positive number and a negative number, then answer is a negative

EXAMPLES:

1) -12 5 = -602) (5)(-4)(-7) = 1403) -1/2(-2/3)(-1) = -1/3 (recall how to multiply fractions)4) (4/5) (-3/10) = -6/25 (recall how to divide fractions)

5) (-3)(-7)|-10| (this expression reads as -3 times -7 times the absolute value of -10)(-3)(-8)(10) = 240 (simplify the absolute value, then multiply)

MORE EXAMPLESThe following examples will include all four operations.

Evaluate each expression:

1) -5 + (-3) - 6 2) -5(-3)(-6) 3) -16 |-10| 4) 15/32 -4/5

5) (-2/5)(-1/3)(5)(-1/2) 6) -2/5 + 1/3 - 5 + 1/2

7) |-7/9| - (-5/6) + 1/3 8) |-11|(-2)|-10|

Answers:

1) -5 + (-3) - 6 = -14 (add/subtract)

2) -5(-3)(-6) = -90 (multiply)

3) -16 |-10| (simplify absolute value)-16 10 = -8/5 or -1 3/5 or -1.6 (divide)

4) 15/32 -4/5 = -3/8 or -0.375 (multiply)

5) (-2/5)(-1/3)(-5)(-1/2) = 1/3 or 0.33 (multiply)

6) -2/5 + 1/3 - 5 + 1/2 = -4 17/30 or -137/30 or - 4.5667 (add/subtract)

7) |-7/9| - (-5/6) + 1/3 (simplify absolute value and opposites)7/9 + 5/6 + 1/3 = 1 17/18 or 35/18 or 1.944 (add/subtract)

8) |-11|(-2)|-10| (simplify absolute value)11 (-2) 10 = -220 (multiply)


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Even and odd numbers

Even and odd numbers are straightforward

concepts. I will start easy, but I will try to

challenge the topic a little bit.

Even numbers

An even number is any number that can bedivided by 2.For example, 12 can be divided by2, so 12 is even.

We saw indivisibility rulesthat a number isdivisible by 2 if its last digit is 0,2,4,6,or 8.

Therefore, any number whose last digit is 0, 2, 4,6, or 8 is an even number

Other examples of even numbers are 58, 44884,998632, 98, 48, and 10000000

Formal definition of an even number:

A number n is even if there exist a number k,such that n = 2k where k is aninteger

This is formal way of saying that if n is divided

by 2, we always get a quotient k with noremainder

Having no remainder means that n can in fact bedivided by 2

Odd numbers

An odd number is any number that cannot bedivided by 2.For example, 25 cannot be dividedby 2, so 25 is odd.

We saw indivisibility rulesthat a number isdivisible by 2 if its last digit is 0,2,4,6,or 8.

Therefore, any number whose last digit is not 0,2, 4, 6, or 8 is an odd number

Other examples of odd numbers are 53, 881,238637, 99, 45, and 100000023

Formal definition of an odd number:A number n is odd if there exist a number k,such that n = 2k + 1 where k is aninteger

This is formal way of saying that if n is dividedby 2, we always get a quotient k with aremainder of 1

Having a remainder of 1 means that n cannot infact be divided by 2

Basic operations with even and odd numbers

Addition

even + even = even

4 + 2 = 6

even + odd = odd

6 + 3 = 9

odd + odd = even

13 + 13 = 26

Multiplication

even even = even

2 6 = 12

even odd = even

8 3 = 24

odd odd = odd

3 5 = 15

Subtraction

even even = even8 4 = 4

even odd = odd

6 3 = 3

odd odd = even13 3 = 10
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Prime Numbers and Composite Numbers

A Prime Number can be divided evenly only by 1 or itself.And it must be a whole number greater than 1.

Example: 7 can only be divided evenly by 1 or 7, so it is a prime number.

But 6 can be divided evenly by 1, 2, 3 and 6 so it is NOT a prime number (it is a compositenumber).

Let me explain ...

Somewhole numberscan be divided up evenly, and some can't!

Example:

6 can be divided evenly by 2, or by 3:

6 = 2 3

Like this:

or

divided into 2 groups divided into 3 groups

But 7 cannot be divided up evenly:

And we give them names:

When a number can be divided up evenly it is a Composite Number When a number can not be divided up evenly it is a Prime Number

So 6 is Composite, but 7 is Prime.
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Not Into Fractions

We are only dealing with whole numbers here! We are not going to cut things into halves orquarters.

Not Into Groups of 1

OK, we could have divided 7 into seven 1s (or one 7) like this:

7 = 1 x 7

But we could do that for any whole number!

So we should also say we are not interested in dividing by 1, or by the number itself.

It is a Prime Number when it can't be divided evenly by any number(except 1 or itself).

Example: is 7 a Prime Number or Composite Number?

You cannot divide 7 evenly by 2 (you would get 2 lots of 3, with one left over) You cannot divide 7 evenly by 3 (you would get 3 lots of 2, with one left over) You cannot divide 7 evenly by 4, or 5, or 6.

You can only divide 7 into one group of 7 (or seven groups of 1):

7 = 1 x 7

So 7 can only be divided evenly by 1 or itself:

So 7 is a Prime Number

And also:

It is a Composite Number when it can be divided evenlyby numbers other than 1 or itself.

Like this:

Example: is 6 a Prime Number or Composite Number?


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6 can be divided evenly by 2, or by 3, as well as by 1 or 6:

6 = 1 66 = 2 3

So 6 is a Composite Number

Sometimes a number can be divided evenly many ways:

Example: 12 can be divided evenly by 1, 2, 3, 4, 6 and 12:

1 12 = 122 6 = 123 4 = 12

So 12 is a Composite Number

And note this:

Any whole number greater than 1 is either Prime or Composite

What About 1?

Years ago 1 was included as a Prime, but now it is not:

1 is neither Prime nor Composite.

Factors

You can make the same definitions using Factors.

"Factors" are the numbers you multiplytogether to get another number.

So here is just a different way of saying the same thing from above:

When the only two factors of a number are 1 and the number,then it is a Prime Number

And remember this is only aboutWhole Numbers(1, 2, 3, ... etc), not fractions or negativenumbers. So don't say "I could multiply times 6 to get 3"OK?

Examples:
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3 = 1 3(the only factors are 1 and 3)

Prime

6 = 1 6 or 6 = 2 3(the factors are 1,2,3 and 6)

Composite

Examples From 1 to 14

I have highlighted any factors other than 1 or the number itself:

NumberCan be Evenly

Divided ByPrime, or

Composite?

1 (1 is not considered prime or composite)

2 1, 2 Prime

3 1, 3 Prime

4 1, 2, 4 Composite

5 1, 5 Prime

6 1, 2, 3, 6 Composite

7 1, 7 Prime

8 1, 2, 4, 8 Composite

9 1, 3, 9 Composite

10 1, 2, 5, 10 Composite

11 1,11 Prime

12 1, 2, 3, 4, 6, 12 Composite

13 1, 13 Prime

14 1, 2, 7, 14 Composite

... ... ...

So when there are more factors than 1 or the number itself, the number is Composite.

A question for you: is 15 Prime or Composite?


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Prime Factorization

Prime Numbers

APrime Numbercan be divided evenly only by 1 or itself.And it must be a whole number greater than 1.

The first few prime numbers are: 2, 3, 5, 7, 11, 13, and 17 ..., and we have aprime number chartif you need more.

Factors

"Factors" are the numbers you multiply together to get another number:

Prime Factorization

"Prime Factorization" is finding which prime numbers multiply together to make the originalnumber.

Here are some examples:

Example 1: What are the prime factors of 12 ?

It is best to start working from the smallest prime number, which is 2, so let's check:

12 2 = 6

Yes, it divided evenly by 2. We have taken the first step!

But 6 is not a prime number, so we need to go further. Let's try 2 again:

6 2 = 3

Yes, that worked also. And 3 is a prime number, so we have the answer:

12 = 2 2 3

As you can see, every factor is a prime number, so the answer must be right.

Note: 12 = 2 2 3 can also be written usingexponentsas 12 = 22 3
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Example 2: What is the prime factorization of 147 ?

Can we divide 147 evenly by 2? No, so we should try the next prime number, 3:

147 3 = 49

Then we try factoring 49, and find that 7 is the smallest prime number that works:

49 7 = 7

And that is as far as we need to go, because all the factors are prime numbers.

147 = 3 7 7

(or 147 = 3 72 using exponents)

Example 3: What is the prime factorization of 17 ?

Hang on ... 17 is a Prime Number.

So that is as far as we can go.

17 = 17

Another Method

We showed you how to do the factorization by starting at the smallest prime and working

upwards.

But sometimes it is easier to break a number down into any factors you can ... then work thosefactor down to primes.

Example: What are the prime factors of 90 ?

Break 90 into 9 10

The prime factors of 9 are 3 and 3

The prime factors of 10 are 2 and 5

So the prime factors of 90 are 3, 3, 2 and 5

Why?

A prime number can only be divided by 1 or itself, so it cannot be factored any further!

Every other whole number can be broken down into prime number factors.


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It is like the Prime Numbers are the basic building blocks of all numbers.

This can be very useful when working with big numbers, such as in Cryptography.

Cryptography

Cryptography is the study of secret codes. Prime Factorization is very important to people whotry to make (or break) secret codes based on numbers.

That is because factoring very large numbers is very hard, and can take computers a long time todo.

If you want to know more, the subject is "encryption" or "cryptography".

Unique

And here is another thing:

There is only one (unique!) set of prime factors for any number.

Example The prime factors of 330 are 2, 3, 5 and 11:

330 = 2 3 5 11

There is no other possible set of prime numbers that can be multiplied to make 330.

In fact this idea is so important it is called theFundamental Theorem of Arithmetic.

Prime Factorization Tool

OK, we have one more method ... use ourPrime Factorization Toolthat can work out the primefactors for numbers up to 4,294,967,296.
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Greatest Common Factor

The highest number that divides exactly into two or more numbers.It is the "greatest" thing for simplifying fractions!

Let's start with an Example ...

Greatest Common Factor of 12 and 16

1. Find all the Factors of each number,

2. Circle the Common factors,

3. Choose the Greatest of those

So ... what is a "Factor" ?

Factors are the numbers you multiply together to get another number:

A number can have many factors:

Factors of 12 are 1, 2, 3, 4, 6 and 12 ...

... because 2 6 = 12, or 4 3 = 12, or 1 12 = 12.

(Read how to findAll the Factors of a Number. In our case we don't need the negative ones.)

What is a "Common Factor" ?

Let us say you have worked out the factors of two numbers:

Example: Factors of 12 and 30

Factors of 12 are 1, 2, 3, 4, 6 and 12

Factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30

Then the common factors are those that are found in both lists:

Notice that 1, 2, 3 and 6 appear in both lists?

So, the common factors of 12 and 30 are: 1, 2, 3 and 6
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It is a common factor when it is a factor of two or more numbers.(It is then "common to"those numbers.)

Here is another example with three numbers:

Example: The common factors of 15, 30 and 105Factors of 15 are 1, 3, 5, and 15



The factors that are common to all three numbers are 1, 3, 5 and 15

In other words, the common factors of 15, 30 and 105 are 1, 3, 5 and 15

What is the "Greatest Common Factor" ?

It is simply the largest of the common factors.

In our previous example, the largest of the common factors is 15, so the Greatest CommonFactor of 15, 30 and 105 is 15

The "Greatest Common Factor" is the largest of the common factors (of two or more numbers)

Why is this Useful?

One of the most useful things is when we want to simplify a fraction: Example: How could wesimplify 12/30?

Earlier we found that the Common Factors of 12 and 30 were 1, 2, 3 and 6, and so the GreatestCommon Factor is 6.So the largest number we can divide both 12 and 30 evenly by is 6, likethis:

6

12

=

2

30 5


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6

The Greatest Common Factor of 12 and 30 is 6.

And so12

/30can be simplified to2

/5

Finding the Greatest Common Factor

1. You can:

find all factors of both numbers (I have anAll Factors Calculatorto help you),

then select the ones that are common to both, and

then choose the greatest.

Example:

Two Numbers Factors Common FactorsGreatest

Common Factor

Example Simplified

Fraction

9 and 129: 1,3,9

12: 1,2,3,4,6,121,3 3 9/12 =

3/4

And another example:

Two Numbers Factors Common FactorsGreatest

Common Factor

Example Simplified

Fraction

6 and 186: 1,2,3,6

18: 1,2,3,6,9,181,2,3,6 6 6/18 =

1/3

2. You can find theprime factorsand combine the common ones together:

Two Numbers Thinking ...Greatest

Common Factor

Example Simplified

Fraction

24 and 1082 2 2 3 = 24, and

2 2 3 3 3 = 1082 2 3 = 12 24/108 =

2/9
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Least Common Multiple

The smallest (non-zero) number that is a multiple of two or more numbers.

Least Common Multiple is made up of the wordsLeast, Common andMultiple:

What is a "Multiple" ?

The multiples of a number are what you get when you multiply it by other numbers (such as ifyou multiply it by 1,2,3,4,5, etc). Just like the multiplication table.

Here are some examples:

The multiples of3 are: 3, 6, 9, 12, 15, 18, 21, etc ...

The multiples of12 are: 12, 24, 36, 48, 60, 72, etc...

What is a "Common Multiple" ?

When you list the multiples of two (or more) numbers, and find the same value in both lists,then that is acommon multiple of those numbers.

For example, when you write down the multiples of4 and 5, the common multiples are those that

are found in both lists:

The multiples of 4 are: 4,8,12,16,20,24,28,32,36,40,44,...

The multiples of 5 are: 5,10,15,20,25,30,35,40,45,50,...

Notice that 20 and 40 appear in both lists?So, the common multiples of 4 and 5 are: 20, 40, (and 60, 80, etc ..., too)

What is the "Least Common Multiple" ?

It is simply the smallest of the common multiples.

In our previous example, the smallest of the common multiples is 20 ...

... so the Least Common Multiple of 4 and 5 is 20.

Finding the Least Common Multiple

It is a really easy thing to do. Just start listing the multiples of the numbers until you get a match.


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Example: Find the least common multiple for 3 and 5:

The multiples of 3 are3, 6, 9, 12, 15, ...,and the multiples of 5 are5, 10, 15, 20, ..., like this:

As you can see on this number line, the first time the multiples match up is 15. Answer: 15

More than 2 Numbers

You can also find the least common multiple of 3 (or more) numbers.

Example: Find the least common multiple for 4, 6, and 8

Multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, 36, ...Multiples of 6 are: 6, 12, 18, 24, 30, 36, ...Multiples of 8 are: 8, 16, 24, 32, 40, ....

So, 24 is the least common multiple (I can't find a smaller one !)

Hint: You can have smaller lists for the bigger numbers.


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Operations on Integers

There are three appealing ways to understand how to add integers. We can usemovement,temperatureandmoney. Lastly, we will take a look at therules for addition.

Movement

You are probably familiar with a number line (seeFigure 1). Traditionally, zero is placedin the center. Positive numbers extend to the right of zero and negative numbers extend tothe left of zero. In order to add positive and negative integers, we will imagine that weare moving along a number line.

Figure 1

ex 1: If asked to add 4 and 3, we would start by moving to the number 4 on the numberline -- exactly four units to the right of zero. Then we would move three units to the right.Since we landed up seven units to the right of zero as a result of these movements, theanswer must be 7.

ex 2: If asked to add 8 and -2, we would start by moving eight units to the right of zero.Then we would move two units left from there because negative numbers make us move

to the left side of the number line. Since our last position is six units to the right of zero,the answer is 6.

ex 3: If asked to add -13 and 4, we start by moving thirteen units to the left of zero. Thenwe move four units to the right. Since we land up nine units to the left of zero, the answeris -9.

ex 4: If asked to add -6 and -5, first move six units to the left of zero. Then move fiveunits further left. Since we are a total of eleven units left of zero, the answer is -11.

Temperature

The temperature model for adding integers is exactly the same as the movement modelbecause most thermometers are really number lines that stand upright. The numbers canbe thought of as temperature changes. Positive numbers make the temperature indicatorrise. Negative numbers make the temperature indicator fall.

Adding two positive temperatures will result in a positive temperature, similar to example1 above. Adding two negative temperatures will result in a negative temperature, similarto example 4 above.
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Examples 2 and 3 can be understood in a different way by imagining a battle betweentwo temperatures. When we added 8 and -2 in example 2, there was more positivetemperature than negative temperature which would explain the result -- positive 6. Inexample 3 there was more negative temperature than positive. That will explain why theanswer is negative.

Money

It can be helpful to think of money when doing integer addition. The positive numbersrepresent income while the negative numbers represent debt.

When adding two incomes, like example 1 above, the answer has to be a bigger incomeand the result is a positive number. When adding two debts, like example 4 above, theanswer has to be another debt. In fact, accountants would call it 'falling deeper in debt.'

Similar to our temperature battle between warm temperatures and cold temperatures,

adding positive and negative numbers is like comparing income to debt. If there is moreincome than debt the answer will be positive, like example 2. If there is more debt thanincome the answer will be negative, like example 3.

Rules for Addition

Below is a table to help condense the rules for addition. Note the second and third rowsof the body of the table. Those answers are dependant upon the original values.

Rules for Addition

Positive + Positive Positive

Positive + Negative Depends

Negative + Positive Depends

SubtractionInstead of coming up with a new method for explaining how to subtract integers, let us borrowfrom the explanation above under the addition of integers. We will learn how to transformsubtraction problems into addition problems.

The technique for changing subtraction problems into addition problems is extremelymechanical. There are two steps:

1. Change the subtraction sign into an addition sign.2. Take the opposite of the number that immediately follows the newly placed addition sign.

Let's take a look at the problem 3 - 4. According to step #1, we have to change the subtractionsign to an addition sign. According to step #2, we have to take the opposite of 4, which is -4.Therefore the problem becomes 3 + (-4). Using the rules for addition, the answer is -1.

Here is another problem: -2 - 8. Switching the problem to an addition problem, it becomes -2 + (-8), which is equal to -10.


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6 - (-20) is equal to 6 + 20, which is 26.-7 - (-1) is the same as -7 + 1, which is -6.

Quizmaster:Subtracting Integers

MultiplicationThe best place to start with multiplication, is with the rules:

Rules for Multiplication

Positive x Positive Positive

Positive x Negative Negative

Negative x Positive Negative

Negative x Negative Positive

Now we have to understand the rules. The first rule is the easiest to remember because we

learned it so long ago. Working with positive numbers under multiplication always yeildspositive answers. However, the last three rules are a bit more challenging to understand.

The second and third steps can be explained simultaneously. This is because numbers can bemultiplied in any order. -3 x 7 has the same answer as 7 x -3, which is always true for allintegers. [This property has a special name in mathematics. It is called the commutativeproperty.] For us, this means the second and third rules are equivalent.

One reason why mathematics has so much value is because its usefulness is derived from itsconsistency. It behaves with strict regularity. This is no accident, mind you. This is quitepurposeful.

Keeping this in mind, let's take a look at Figure 2 below. There is a definite pattern to theproblems in the table. The first number in each row remains constant but the second number isdecreasing by one, each step down the table. Consequently, the answer is changing. The answershave a definite pattern as we go down the table too. It should be relatively easy to determine thetwo missing answers.

If you understand the pattern, you will see that the first unanswered problem is -2 and the secondunanswered problem is -4. This should provide some meaning why a negative number is alwaysthe result when multiplying two numbers of opposite sign.

Likewise, lets turn our attention toFigure 3below. This table has a pattern similar to the one inFigure 1. However, this table begins with a negative number. As we scan the list of answers, wecan see that the last two problems remain unanswered.

With a little concentration, we can see that the two unanswered questions must have positiveanswers to maintain mathematical consistency. This should help us understand why a positivenumber is always the result of multiplying two numbers of the same sign.
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Exponents

The exponent of a number says how many times to use the number in amultiplication.

In 82 the "2" says to use 8 twice in a multiplication,so 8

2= 8 8 = 64

Exponents are also called Powers or Indices.

In words: 82

could be called "8 to the power 2" or "8 to the second power", or simply "8squared"

Some more examples:

Example: 53 = 5 5 5 = 125

In words: 53 could be called "5 to the third power", "5 to the power 3" or simply "5cubed"

Example: 24 = 2 2 2 2 = 16

In words: 24 could be called "2 to the fourth power" or "2 to the power 4" or simply "2 tothe 4th"

Exponents make it easier to write and use many multiplications

Example: 96 is easier to write and read than 9 9 9 9 9 9

You can multiply any number by itselfas many times as you want using exponents.

Try here:


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In General

So, in general:

an tells you to multiply a by itself,so there are n of those a's:

Other Way of Writing It

Sometimes people use the ^ symbol (just above the 6 on your keyboard), because it is easy totype.

Example: 2^4 is the same as 24

2^4 = 2 2 2 2 = 16

Negative Exponents

Negative? What could be the opposite of multiplying?

Dividing!

A negative exponent means how many times to divide one by the number.

Example: 8-1 = 1 8 = 0.125

You can have many divides:

Example: 5-3 = 1 5 5 5 = 0.008

But that can be done an easier way:

5-3 could also be calculated like:

1 (5 5 5) = 1/53

= 1/125 = 0.008

In General

That last example showed an easier way to handle negative exponents:

Calculate the positive exponent (an)

Then take theReciprocal(i.e. 1/an)
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More Examples:

Negative Exponent Reciprocal of Positive Exponent Answer

4- = 1 / 4 = 1/16 = 0.0625

10-3 = 1 / 103 = 1/1,000 = 0.001

(-2)- = 1 / (-2) = 1/(-8) = -0.125

What if the Exponent is 1, or 0?

1 If the exponent is 1, then you just have the number itself (example 9 = 9)

0 If the exponent is 0, then you get 1 (example 9 = 1)

But what about 0 ? It could be either 1 or 0, and so people say it is "indeterminate".

It All Makes Sense

My favorite method is to start with "1" and then multiply or divide as many times as theexponent says, then you will get the right answer, for example:

Example: Powers of 5

.. etc..

5 1 5 5 25

5

1

1 5 55 1 1

5- 1 5 0.2

5- 1 5 5 0.04

.. etc..

If you look at that table, you will see that positive, zero or negative exponents are really part ofthe same (fairly simple) pattern.

Be Careful About Grouping

To avoid confusion, use parentheses () in cases like this:

With () : (-2) = (-2) (-2) = 4

Without () : -2 = -(2 ) = - (2 2) = -4

With () : (ab) = ab ab

Without () : ab = a (b) = a b b


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Fractions

Types of Fractions

An Improper fraction has a top numberlarger than (or equal to) the bottom number,

It is "top-heavy"

/4

(seven-fourths or seven-quarters)

Examples

/2 /41 /15

1 /15 /5

See how the top number is bigger than (or equal to) the bottom number?That makes it an Improper Fraction, (but there isnothing wrong about Improper Fractions).

Three Types of Fractions

There are three types of fraction:

Fractions

A Fraction (such as 7/4) has two numbers:

Numerator

Denominator

The top number is the Numerator, it is the number ofparts you have.The bottom number is the Denominator, it is the number ofparts the whole is divided into.

Example: 7/4 means:
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We have 7 parts Each part is a quarter (

1/4) of a whole

So we can define the three types of fractions like this:

Proper Fractions: The numerator is less than the denominatorExamples: 1/3, /4, /7

Improper Fractions: The numerator is greater than (or equal to) the denominator

Examples:4/3,

11/4, /7

Mixed Fractions: A whole number and proper fraction together

Examples: 1 /3, 2 /4, 16 /5

Improper FractionSo, an improper fraction is just a fraction where the top number (numerator) is greater than orequal to the bottom number (denominator).

In other words, it is top-heavy.

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Can be Equal

What about when the numerator is equal to the denominator? For

example

4

/4 ?

Well, it is obviously the same as a whole, but it is written as afraction, so most people agree it is a type of improper fraction.

Improper Fractions or Mixed Fractions

You can use either an improper fraction or a mixed fraction to show the same amount. Forexample 1 3/4 =

7/4, shown here:

1 /4 /4

=


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Converting Improper Fractions to Mixed Fractions

To convert an improper fraction to a mixed fraction, follow these steps:

Divide the numerator by the denominator. Write down the whole number answer Then write down any remainder above the denominator.

Example: Convert 11/4 to a mixed fraction.

Divide:

11 4 = 2 with a remainder of 3

Write down the 2 and then write down the remainder (3) above the denominator (4), like this:

23

4

Converting Mixed Fractions to Improper Fractions

To convert a mixed fraction to an improper fraction, follow these steps:

Multiply the whole number part by the fraction's denominator. Add that to the numerator Then write the result on top of the denominator.

Example: Convert 3 2/5 to an improper fraction.

Multiply the whole number by the denominator:

3 5 = 15

Add the numerator to that: 15 + 2 = 17

Then write that down above the denominator, like this:

17

5


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Fraction Operations

To add (or subtract) twofractions:

1) Find theleast common denominator.

2) Write both original fractions asequivalent fractionswith the least commondenominator.

3) Add (or subtract) the numerators.

4) Write the result with the denominator.

Example 1:

Add .

The least common denominator is 21.

To multiply two fractions:

1) Multiply the numerator by the numerator.

2) Multiply the denominator by the denominator.

For all real numbers a, b, c, d(b 0, d 0)

Example 2:

Multiply .
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To divide by a fraction, multiply by itsreciprocal.

For all real numbers a, b, c, d(b 0, c 0, d 0)

Example 3:

Divide .

Mixed numberscan be written as animproper fractionand an improper fraction can be written as

a mixed number.

Example 4:

Write as an improper fraction.
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Example 5:

Write as a mixed number in simple form.

A fraction is inlowest termswhen the numerator and denominator have no common factor otherthan 1. To write a fraction in lowest terms, divide the numerator and denominator by thegreatestcommon factor.

Example 6:

Write in lowest terms.

45 and 75 have a common factor of 15.
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Decimal Numbers

Any number can be written in "decimal form".

There are three different types of decimal number: exact, recurring and other decimals.

An exactor terminating decimal is one which does notgo on forever, so you can write down allits digits. For example: 0.125

A recurring decimal is a decimal number which does go on forever, but where some of the digitsare repeated over and over again. For example: 0.1252525252525252525... is a recurringdecimal, where '25' is repeated forever.

Sometimes recurring decimals are written with a bar over the digits which are repeated, or withdots over the first and last digits that are repeated.

For example:

Other decimals are those which go on forever and don't have digits which repeat. For example pi= 3.141592653589793238462643...

Relationship with Fractions

In decimal form, arational number(fraction) is either an exact or a recurring decimal.

The reverse is also true: exact and recurring decimals can be written as fractions. For example,0.175 =175/1000 = 7/40. Also, 0.2222222222... is rational since it is a recurring decimal = 2/9.

You can tell if a fraction will be an exact or a recurring decimal as follows: fractions withdenominators that have only prime factors of 2 and 5 will be exact decimals. Others will berecurring decimals. This means that when you write the denominator of a fraction in itsprimefactor decomposition, if there are only 2's and 5's you will get an exact decimal.For example, 1/8. The denominator is 8, which is 2 2 2. There are only 2's in the prime factordecomposition so the decimal will be exact (and it is 0.125).On the other hand, 2/9 has denominator 9 = 3 3 and 3 isn't a 2 or a 5 so we have a recurringdecimal (0.222222....).

Converting a Recurring Decimal to a Fraction

We know that recurring decimals can be written as fractions. The trick is to use a little algebra.

Example

Convert 0.142857142857... into a fraction.

Let x = 0.142857142857...We want to move the decimal point to the right, so that the first "block" of repeated digits
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appears before the decimal point. Remember that multiplying by 10 moves the decimal point 1position to the right.

So in this example, we need to move the decimal point 6 places to the right (so we multiply bothsides by 1 000 000):

1000000x = 142857.142857142857...

Now we can subtract our original number, x, from both sides to get rid of everything after thedecimal point on the right:

1000000x - x = 142857

So 999999x = 142857

x = 142857/999999

= 1/7 (cancelling)

Rounding Numbers

If the answer to a question was 0.00256023164, you would not usually write this down. Instead,you would 'round off' the answer to save space and time. There are two ways to do this: you canround off to a certain number of decimal places or a certain number of significant figures.

0.00256023164, rounded off to 5 decimal places (d.p.) is 0.00256 . You write down the 5numbers after the decimal point. To round the number to 5 significant figures, you write down 5

numbers. However, you do not count any zeros at the beginning. So to 5 s.f. (significant figures),the number is 0.0025602 (5 numbers after the first non-zero number appears).

From what I have just said, if you rounded 4.909 to 2 decimal places, the answer would be 4.90 .However, the number is closer to 4.91 than 4.90, because the next number is a 9. Therefore, therule is: if the number after the place you stop is 5 or above, you add one to the last number youwrite.So 3.486 to 3s.f. is 3.490.0096 to 3d.p. is 0.010 (This is because you add 1 to the 9, making it 10. When rounding to anumber of decimal places, always write any zeros at the end of the number).
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Ratios and Proportions

Knowing how to work with ratios and proportions is very handy in chemistry classes, especiallywhen working with different units of measurement. Let's start with some dictionary definitions:

Ratio: The relative size of two quantities expressed as the quotient of one divided by the other;the ratio of a to b is written as a:b or a/b.Proportion: An equality between two ratios.

So what are these things really? Consider the following situation.

In 1995, 78 women were enrolled in chemistry at a certain high school while 162 men wereenrolled. What was the ratio of women to men? Men to women?

Let's answer the questions using the definition of ratio. Filling in what we know:

women : men is 78:162 or 78/162men : women is 162:78 or 162/78

We could have reduced the fractions (cancelling out a factor of 6) or used our calculators to get adecimal equivalent for these fractions using the divide key:

women : men is 78 162 or 13 27 or 0.481481481men : women is 162 78 or 27 13 or 2.07692308

By writing the answer in these ways we have lost information, namely the specific number ofmen and women. Be careful! When given a ratio such as 13:27, the fractions may have beenreduced so the original quantities could have been larger. This brings us to the idea of proportion.

Ratios are said to be in proportion when their corresponding fractions are equal. What we reallydid above was notice that the fraction 78/162 was equal to the fraction 13/27 - because we coulddivide both numbers in the first ratio by six (6) to get the second ratio - so the ratios are equal aswell, i.e.,

78/162 = 13/27,

or using the colon form 78 : 162 = 13 : 27.

These two (2) equalities are examples of proportions (equal ratios); it is as simple as that. Howare proportions used? Let's add another question to our problem:

In 1996 the number of men enrolled was 193 while the ratio of women to men enrolled inchemistry stayed the about same as in 1995. How many women were enrolled in chemistry in1996?

To answer this, we build a proportion equating the ratio of women to men in the two (2) years:


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78 : 162 = ?? : 193

These are easiest to solve when written in fraction form. Notice that we've used x for theunknown number of women since it is common in algebra to use the letter x as the variable.

78

162=

x

193

We can then use the "cross-multiply" technique:

78 * 193 = 162 xmultiply on our calculator:

15054 = 162 xand then divide on our calculator:

92.9259259 = x

So the number of women enrolled in 1996 was 92.9259259??? Use your common sense! Thereare no fractions of women walking around, so we will report 93 (rounding 92.9259259correctly). It is probably wise to say "There are about 93 women enrolled in chemistry in 1996"since we were told the proportions were about the same. The issue of when to round will comeup in chemistry, too. Often the objects being counted will be atoms or atomic particles likeprotons, neutrons and electrons; only whole number answers make sense in this case. Sometimes,however, we're measuring amounts (such as masses) which can be fractional. Then you shoulduse the rules for correct rounding and significant figures reviewed in Session 1.


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Angles

An angle measures the amount of turn

Names of Angles

As the Angle Increases, the Name Changes

Type of Angle Description

Acute Angle an angle that is less than 90

Right Angle an angle that is 90 exactly

Obtuse Angle an angle that is greater than 90 but less than180

Straight Angle an angle that is 180 exactly

Reflex Angle an angle that is greater than 180

Try It Yourself!

View Larger
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In One Diagram

This diagram might make it easier to remember:

Also: Acute, Obtuse and Reflex are in alphabetical order.

Be Careful What You Measure

This is an Obtuse Angle. And this is a Reflex Angle.

But the lines are the same ... so when naming the angles make surethat you know which angle is being asked for!

Parts of an Angle

The corner point of an angle is called the vertex

And the two straight sides are called arms

The angle is the amount of turn between each arm.

Labelling Angles


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There are two main ways to label angles:

1. by giving the angle a name, usually a lower-case letter like

a or b, or sometimes a Greek letter like (alpha) or (theta)

2. or by the three letters on the shape that define the angle,with the middle letter being where the angle actually is (itsvertex).

Example angle "a" is "BAC", and angle "" is "BCD"

PolygonsA polygon is aplaneshape with straight sides.

Is it a Polygon?

Polygons are 2-dimensional shapes. They are made of straight lines, and the shape is "closed"(all the lines connect up).

Polygon(straight sides)

Not a Polygon(has a curve)

Not a Polygon(open, not closed)

Polygon comes from Greek. Poly- means "many" and -gon means "angle".

Types of Polygons

Simple or Complex

A simple polygon has only one boundary, and it doesn't cross over itself. A complex polygonintersects itself! Many rules about polygons don't work when it is complex.
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Simple Polygon(this one's a Pentagon)

Complex Polygon(also a Pentagon)

Concave or Convex

A convex polygon has no angles pointing inwards. More precisely, no internal angles can bemore than 180.

If there are any internal angles greater than 180 then it is concave. (Think: concave has a "cave"in it)

Convex Concave

Regular or Irregular

If all angles are equal and all sides are equal, then it is regular, otherwise it is irregular

Regular Irregular

More Examples

Complex Polygon(a "star polygon", in

this case, apentagram)

Concave OctagonIrregular Hexagon
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Play With Them!

TryInteractive Polygons... make them concave, complex or regular.

Names of Polygons

If it is a Regular Polygon...

Name Sides Shape Interior Angle

Triangle(or Trigon) 3 60

Quadrilateral(or Tetragon) 4 90

Pentagon 5 108

Hexagon 6 120

Heptagon (or Septagon) 7 128.571

Octagon 8 135

Nonagon (or Enneagon) 9 140

Decagon 10 144

Hendecagon (or Undecagon) 11 147.273

Dodecagon 12 150

Triskaidecagon 13 152.308

Tetrakaidecagon 14 154.286

Pentadecagon 15 156

Hexakaidecagon 16 157.5

Heptadecagon 17 158.824

Octakaidecagon 18 160

Enneadecagon 19 161.053

Icosagon 20 162

Triacontagon 30 168
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Tetracontagon 40 171

Pentacontagon 50 172.8

Hexacontagon 60 174

Heptacontagon 70 174.857

Octacontagon 80 175.5Enneacontagon 90 176

Hectagon 100 176.4

Chiliagon 1,000 179.64

Myriagon 10,000 179.964

Megagon 1,000,000 ~180

Googolgon 10100 ~180

n-gon n (n-2) 180 /n

For polygons with 13 or more sides, it is OK (and easier) to write "13-gon","14-gon" ... "100-gon", etc.


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Mathematical Symbols

Below, find a comprehensive list of most basic mathematical symbols used in basic math

Symbols Meaning Symbols Meaning

=

+

A

m

S.A.

L.A

B

square root

less than

greater than

not equal

equal

equivalent

approximately

smaller or equal

bigger or equal

division

multiplication

addition

subtraction

angle

degree

pi (3.14)

area

slope of a line

surface area

lateral area

area of base

%

GCF

LCM

|

a : b

an

||

| |

()

b

h

p or P

l

w

C

-a

d

b1, b2

fraction bar

right angle sign

percent sign

plus or minus sign

greatest common factor

least common multiple

divides

ratio

a to the nth power

parallel lines

sign for absolute value

parentheses for grouping

base length

height

perimeter

Length or slant height

width

circumference

opposite of a

diameter or distance

base lengths of a trapezoid


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V

ABC

volume

perpendicular

triangle ABC

r

ABC

mABC

rate or radius

angle ABC

refers to the measure of angle ABC

Linear Equation in One Variable

Basic Algebra > Linear Equation in One Variable

Linear Equation in One Variable

1. Definition of an Equation:

What is an equation?

A statement of equality of two algebraic expressions in or more variables is called an equation.

Examples: 1. x + 1 = 2 and

2. 2y + 3 = 5

In 1 above, x + 1 is an algebraic expression in the variable x. we read it as x plus 1 is equal to

2.

x is some number, called variable.

In 2 above, 2y + 3 is an algebraic expression in the variable y, we read it as 2 times y or 2y plus

3 is equal to 5. y is some number, called variable

2. The three components of an equation

Every equation has a Left hand side, the equality sign = and the Right hand side. the three

components in the equation x + 1 = 2 are :

L.H.S = Left Hand Side

R.H.S = Right Hand Side

3. Solution/Root of an Equation:


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the value of x, i.e. some number for x, which makes the equation a true statement is called

solution or root of the equation.

In simple words, if the L.H.S. and R.H.S become equal for some number plugged in for x, then

the number, also called value, is the solution or root of the equation.

In x + 1 = 2, what should be plugged in for x so that L.H.S. becomes equal to R.H.S?

It is 1, i.e. if 1 is plugged in for x, the two sides become equal.

This number or value 1 for x is called root or solution of the equation. Transposition Rule:

Transposition Rule

1. The transposition rule applies on addition and subtraction.

Transposition Rule

Terms can be transposed (shifted) between either sides of the equality symbol = with a

change in sign of the transposed terms

Example:x + 1 = 2.

1 can be transposed to right side by inversing its sign

i.e. x = -1 + 2 so that we have x = 1

transposition conforms with the rule:

same numbers can be added on both sides of an equation

So, in x + 1 = 2, to find x, we need to get rid of 1 on the left hand side, to do this add -1 on both

sides of the equation.

x +1-1 = -1 + 2,

x = 1

Application of Linear Equations or Word Problems on Linear Equations in one Variable

Problem 1:

The sum of two consecutive numbers is 25. Find the numbers.

Solution:

Let the two consecutive numbers be x andx+1.

So we can set up the following linear equation:

Given that x + x+1= 25,

2x = 24, {from transposition rule, inverse sign of 1 on taking it to right side}

x = 24/2 = 12, {multiplication changes to division}


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So the other number is x + 1 = 12 + 1 = 13

Therefore, the two numbers are 12 and 13

Problem 2:

John is 15 years now. He is 10 years older than his brother Tom. How old is Tom 10 years

from now?

Solution: Let Tom be x years old now.

So, we set up the linear equation:

x + 10 = 15, x = 5. {from transposition rule, inverse sign of 10 on taking it to right side}

So, tom is 5 years old now.

10 years from now, tom will be 10 + 5 = 15

Problem 3:

20 years from now, Nancy will become three times as old as he is now. Find her age now.

Solution:

Let Nancy be x years old now.

20 years from now, she will be x +20

But 20 years from now, she we thrice her present age, x i.e. 3x

So, x + 20 = 3x, or 3x = x + 20,

3x x = 20, {from transposition rule}

2x = 20,

x = 20/2 = 10

Linear Inequality

Definition of Linear Inequality

A Linear Inequality involves a linear expression in two variables by using any of therelational symbols such as , or

More about Linear Inequality

A linear inequality divides a plane into two parts. If the boundary line is solid, then the linear inequality must be either or . If the boundary line is dotted, then the linear inequality must be either > or


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As theboundary line in the above graph is a solid line, the inequality must be either or. Since the region below the line is shaded, the inequality should be . We can noticethat the line y = - 2x + 4 is included in the graph; therefore, the inequality is y - 2x + 4.Any point in the shaded plane is a solution and even the points that fall on the line arealso solutions to the inequality.

4x + 6y 12, x + 6 14, 2x - 6y < 12 + 2x, 9y < 12 + 2x are the examples of linearinequalities.

Solved Example on Linear Inequality

Which of the graphs best suits the inequality y < x - 4?

Choices:A. Graph 1B. Graph 2C. Graph 3


65/125

D. Graph 4Correct Answer: ASolution:Step 1: Since the inequ

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